First-order formulae are concise in acylindrically hyperbolic groups and other classes including Burnside, ICC, and torus knot groups, with results on finiteness of definable sets.
Baumslag,Some aspects of groups with unique roots, Acta Math.104(1960), 217–303
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Concise formulae in groups of non-positive curvature
First-order formulae are concise in acylindrically hyperbolic groups and other classes including Burnside, ICC, and torus knot groups, with results on finiteness of definable sets.